Modelling and analysis of the influence of in-plane vertical modes on the internal resonance of cable-stayed bridges

Abstract

To study the complex mechanism of internal resonance in cable-stayed bridges, an integrated dynamic system comprised of four stay cables and a bridge beam was established in this study. Considering the effect of the beam’s shear force and axial force and by applying the finite difference method, the dynamic equations under non-linear boundary conditions were derived based on the D’Alembert principle and Galerkin method. By comparing the results obtained from the dynamic equations with those obtained by a finite element model (FEM), the linearization and non-linear characteristics of the dynamic model were further verified and discussed. The results of the analytical simulation based on the 4th Runge–Kutta Method show that a severe internal resonance is observed once the ratio of local mode and the undamped in-plane vertical natural mode (IVNM) meets 1:1 or 1:2. The conditions regarding the threshold of the initial excitations are also discussed. For the exceptions, the concept of the zero-point of a mode shape (ZPS), where the correlation coefficient in the vertical mode of a certain order equals zero, was proposed and applied in the analysis of the influence of the IVNMs on internal resonance. It is further illustrated that the cables located at the ZPS are not excited by the IVNM of the corresponding order. Additionally, when the excitations act on the ZPS, the IVNMs of the corresponding order are not initially observed in the structure. By using the filtering technology with a zero-phase-shift, the separated vibration signals further revealed that the IVNM becomes re-excited in the global system and thus the re-excited resonance occurs in the cables under specific conditions: an increase in time and excitation value. Three conditions regarding the modal interaction processes of the re-excited resonance were further investigated and discussed.